Smart quantum light detector

ABSTRACT

A method and system for identification of light source types includes detecting individual photons for a measurement time period to provide a times series of individual photon events, segmenting the time series into a plurality of time bins, and determining a number of detected photons within each time bin to provide a time series of photon counts, determining a probability distribution P(n) from the time series of photon counts, the probability distribution providing the probability of detection of n photons (n=0 . . . n max ), inputting each of the values of P(n) as a n max +1 component feature vector into a single neuron neural network that has been previously trained on a plurality of light source types, and receiving as output a classifier that has a value that identifies the light source type. An average number of photons in the plurality of time bins is less than one photon.

CROSS REFERENCE TO RELATED APPLICATIONS

The present patent application claims priority benefit to U.S.Provisional Application No. 63/079,290 filed on Sep. 16, 2020, theentire content of which is incorporated herein by reference. Allreferences cited anywhere in this specification, including thebackground and detailed description sections, are incorporated byreference as if each had been individually incorporated.

BACKGROUND 1. Technical Field

The presently claimed embodiments of the current invention relate tolight detectors, methods of light detection and systems that includesuch detectors and/or methods; and more particularly to such detectors,methods and systems that use neural networks for characterization oflight sources.

2. Discussion of Related Art

The underlying statistical fluctuations of the electromagnetic fieldhave been widely utilized to identify diverse sources of light. In thisregard, the Mandel parameter constitutes an important metric tocharacterize the excitation mode of the electromagnetic field andconsequently to classify light sources. Similarly, the degree of opticalcoherence has also been extensively utilized to identify light sources.Despite the fundamental importance of these quantities, they requirelarge amounts of data, which impose practical limitations. This problemhas been partially alleviated by incorporating statistical methods, suchas bootstrapping, to predict unlikely events that are hard to measureexperimentally. Unfortunately, the constraints of these methods severelyimpact the realistic implementation of photonic technologies formetrology, imaging, remote sensing, and microscopy.

The potential of machine learning (ML) has motivated novel families oftechnologies that exploit self-learning and self-evolving features ofartificial neural networks to solve a large variety of problems indifferent branches of science. Conversely, quantum mechanical systemshave provided new mechanisms to achieve quantum speedup in machinelearning. In the context of quantum optics, there has been an enormousinterest in utilizing machine learning to optimize quantum resources inoptical systems. As a tool to characterize quantum systems, machinelearning has been successfully employed to reduce the number ofmeasurements required to perform quantum state discrimination, quantumseparability, and quantum state tomography.

However, there remains a need to perform discrimination of light sourcesat extremely low light levels.

SUMMARY OF THE DISCLOSURE

An aspect of the present invention is to provide a method foridentification of light source types. The method includes detectingindividual photons for a measurement time period to provide a timesseries of individual photon events. The method further includessegmenting the time series into a plurality of time bins, anddetermining a number of detected photons within each time bin of theplurality of time bins to provide a time series of photon counts pertime bin. The method also includes determining a probabilitydistribution P(n) from the time series of photon counts per time bin,the probability distribution providing the probability of detection of nphotons, wherein n=0, 1, 2, . . . , n_(max), inputting each of thevalues of P(n) as a n_(max)+1 component feature vector into a singleneuron neural network, the single neuron neural network having beenpreviously trained on a plurality of light source types, and receivingas output a classifier that has a value that identifies the light sourcetype. An average number of photons in the plurality of time bins is lessthan one photon.

Another aspect of the present disclosure is to provide a light detectionsystem for detecting light from a classified type of light source. Thelight detection system includes a light detector, and a processingsystem configured to communicate with said light detector to receivesignals to be processed. The processing system is constructed to performthe method for identification of light source types in the aboveparagraph.

A further aspect of the present invention is to provide an opticalimaging system for forming images from a classified type of lightsource. The optical imaging system includes a plurality of lightdetectors arranged in a patterned array; and a processing systemconfigured to communicate with said plurality light detectors to receivesignals to be processed to provide an image from said classified type oflight source. The processing system is constructed to perform the methodfor identification of light source types in the above paragraph.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure, as well as the methods of operation andfunctions of the related elements of structure and the combination ofparts and economies of manufacture, will become more apparent uponconsideration of the following description and the appended claims withreference to the accompanying drawings, all of which form a part of thisspecification, wherein like reference numerals designate correspondingparts in the various figures. It is to be expressly understood, however,that the drawings are for the purpose of illustration and descriptiononly and are not intended as a definition of the limits of theinvention. All references cited anywhere in this specification,including the Background and Detailed Description sections, areincorporated by reference as if each had been individually incorporated.

FIG. 1 is a flow diagram showing the structure of ADAptive LINearElement (ADALINE) model, according to an embodiment of the presentinvention;

FIG. 2 is a schematic diagram of a smart quantum camera for remotesensing, according to an embodiment of the present invention;

FIG. 3 is a schematic representation of an experimental apparatus forusing coherent beam of light and thermal beam of light, according to anembodiment of the present invention;

FIGS. 4A-4D shows a set of histograms displaying theoretical andexperimental photon number probability distributions for coherent andthermal light beams with different mean photon numbers, according to anembodiment of the present invention;

FIGS. 5A-5B show the probability distributions of coherent and thermallight, for varying dataset sizes (10, 20, 50, 100, 10 k), according toan embodiment of the present invention;

FIG. 6 is a plot of the overall accuracy of light discrimination versusthe number of data points used in naive Bayes classifier;

FIG. 7 is a plot of the overall accuracy of light discrimination versusthe number of data points used in ADALINE, according to an embodiment ofthe present invention;

FIGS. 8A-8D show a 3-dimensional (3D) projection of the feature space onthe plane (P(0), P(1), P(2)) for different mean photon numbers,according to an embodiment of the present invention;

FIGS. 9A-9D shows 3D Projection of the feature space on the plane (P(0),P(1), P(2)) for different number of data points, according to anembodiment of the present invention;

FIG. 10A is a schematic structure representation one-dimensionalconvolutional neural network 1D-CNN), according to an embodiment of thepresent invention;

FIG. 10B is a schematic structure representation of a multilayer neuralnetwork (MNN) used for demonstration of light source identification,according to another embodiment of the present invention;

FIGS. 11A-11B are plots of the overall accuracy of light discriminationversus the number of neurons in the hidden layer of the MNN byconsidering two different mean photon numbers, according to anembodiment of the present invention;

FIGS. 12A-12B are plots of the overall accuracy of light discriminationversus the number of data points, according an embodiment of the presentinvention;

FIGS. 13A and 13B is a conceptual illustration and schematic of anotherexperimental setup to demonstrate super-resolving imaging, according toan embodiment of the present invention;

FIG. 14A shows a scheme of the two-layer neural network used to identifythe photon statistics produced by a combination of three sources,according to an embodiment of the present invention;

FIG. 14B shows the performance of the present neural network as afunction of the number of data samples used each time in the testingprocess, according to an embodiment of the present invention;

FIGS. 15A-15F show various aspects of the experimental super-resolvingimaging, according to an embodiment of the present invention; and

FIG. 16 shows a comparison between the spatial resolution of our cameraand direct imaging, according to an embodiment of the present invention.

DETAILED DESCRIPTION

Some embodiments of the current invention are discussed in detail below.In describing embodiments, specific terminology is employed for the sakeof clarity. However, the invention is not intended to be limited to thespecific terminology so selected. A person skilled in the relevant artwill recognize that other equivalent components can be employed, andother methods developed, without departing from the broad concepts ofthe present invention. All references cited anywhere in thisspecification are incorporated by reference as if each had beenindividually incorporated.

As used herein, the term “light” is intended to have a broad meaning toregions of the electromagnetic spectrum that are both visible and notvisible to the human eye. For example, the term light is intended toinclude, but is not limited to, visible light, infrared light (IR) andultraviolet light (UV).

According to some embodiments of the current invention, we demonstratethe potential of machine learning (ML) to perform discrimination oflight sources at extremely low light levels. This is achieved, accordingto an embodiment of the current invention, by training single artificialneurons with the statistical fluctuations that characterize coherent andthermal states of light. The self-learning features of artificialneurons enable the dramatic reduction in the number of measurements andthe number of photons required to perform identification of lightsources. For the first time, our results demonstrate the possibility ofusing tens of measurements to identify light sources with mean photonnumbers below one according to an embodiment of the current invention.In addition, we demonstrate similar experimental results using the naiveBayes classifier, which are outperformed by our single neuron approach.Finally, we present a discussion on how a single artificial neuron basedon an ADAptive LINear Element (ADALINE) model can dramatically reducethe number of measurements required to discriminate signal photons fromambient photons. Some embodiments of the current invention can provide,for example, realistic implementation of light detection and ranging(LiDAR), remote sensing, and microscopy. However, the broad concepts ofthe current invention are not limited to only these particular examples.

In order to dramatically reduce the number of measurements required toidentify light sources, we can make use of an ADALINE neuron accordingto an embodiment of the current invention. ADALINE is a single neuralnetwork model based on a linear processing element, proposed by BernardWidrow for binary classification. In general, the neural networksundergo two-stage: training and test. In the training stage, ADALINE iscapable of learning the correct outputs (named as output labels orclasses) from a set of inputs, called also features, by using asupervised learning algorithm. In the test stage, this neuron producesthe outputs of a set of inputs that were not in the training data,taking as reference the acquired experience in the training stage.Although we tested architectures more complex than a single neuron forthe identification of light sources, we concluded that a simple ADALINEoffers a suitable agreement between accuracy and simplicity.Furthermore, the training time is insignificantly small.

FIG. 1 is a flow diagram showing the structure of the ADALINE model,according to an embodiment of the present invention. In FIG. 1 , P(n)denotes the probability of finding n photons for a given light source,namely, coherent or thermal. We take as an input to the neuron, thefeature vector composed by the first seven probabilities of the photonnumber distribution, that is, P={P(0), P(1), P(2), P(3), P(4), P(5),P(6)}. With this, we achieve that the feature vector size stays fixedfor a different number of data points. It is worth mentioning that thedetermination of an appropriate feature vector is one hard task inmachine learning. Note however, that the general concepts of the currentinvention are not limited to a seven component feature vector asdescribed in this example. There can be less than seven components, orgreater than seven components in other embodiments of the currentinvention.

ADALINE infers a function from the set of training examples, which afterit is used to predict output labels of new input data. The neuron'soutput is given by the following equation set (1).

a=f(z)z=Σ _(i)ω_(i) x _(i)   (1)

where x_(i) (i=0, . . ., 7) are the elements of the feature vector P. x₀is a bias term and is permanently set to 1. ω_(i) are the synapticweights associated to each input where ω₀ corresponds to the weight ofthe bias and f (⋅) is the identity activation function which takes formof f(x)=x.

We note that the output of the activation function undergoes a binaryclassification given by the threshold function, there, if a is greateror equal to 0.5 then the output belongs to the class labeled ascoherent, whereas, if a<0, the output belongs to the thermal class.Importantly, these two classes are a consequence of adjusting theweights defining the hyper-plane equation given by z=0 (also calleddecision surface), due to that the hyper plane divides into two regionsthe feature space. Thus, each possible input is assigned to one of thetwo regions. In the training stage, the weights are initially set torandom values. After each observation (input), they are updatedfollowing a learning rule referred to as the delta rule given byequation (2):

ω_(i)(k+1)=ω_(i)(k)+ηE(k)x _(i)(k)   (2)

where k is a particular observation and i is a constant known as thelearning rate. E(k) is the resulting error between the target output andneuron's output at k-th observation. Equation (2) can be derived of thegradient descent method taking as a cost function the mean squarederror.

Accordingly, an embodiment of the current invention is directed to amethod for identification of light source types. The method includesdetecting individual photons for a measurement time period to provide atimes series of individual photon events. The method further includessegmenting the time series into a plurality of time bins, anddetermining a number of detected photons within each time bin of theplurality of time bins to provide a time series of photon counts pertime bin. The method also includes determining a probabilitydistribution P(n) from the time series of photon counts per time bin,where the probability distribution provides the probability of detectionof n photons (n=0, 1, 2, . . . , n_(max)), inputting each of the valuesof P(n) as a n_(max)+1 component feature vector into a single neuronneural network, the single neuron neural network having been previouslytrained on a plurality of light source types, and receiving as output aclassifier that has a value that identifies the light source type. Theaverage number of photons in the plurality of time bins can be less thanone photon.

In some embodiments, the light source type is one of a coherent lightsource or a thermal light source. In some embodiments, n_(max) is equalto 6 and the feature vector is a seven-component feature vector. In someembodiments, the single neuron neural network includes an identityactivation function and a binary classification given by a thresholdfunction to indicate a class labeled as coherent on a first side of athreshold or a class labeled thermal on a second side of the threshold.

In some embodiments, the plurality of time bins is less than 100. Insome embodiments, the plurality of time bins is less than 20. In someembodiments, the plurality of time bins each have substantially equaltemporal widths and have a value selected to correspond to a coherencetime of the coherent light source. In some embodiments, the methodfurther includes training the single neuron neural network prior to theidentifying the light source type.

A light detection system for detecting light from a classified type oflight source according to an embodiment of the current inventionincludes a light detector and a processing system that is configured tocommunicate with the light detector to receive signals to be processed.The processing system is constructed to perform any one of theabove-noted methods according to embodiments of the current invention.

Another embodiment of the current invention is directed to a new familyof quantum cameras or imaging systems endowed with the capability ofidentifying sources of light at each pixel. This technology can haveenormous implications for microscopy, remote sensing, and astronomy.Embodiments of a smart quantum detector that enable the identificationof light sources at the single-photon level are described above. Thiscan be used to exploit quantum fluctuations of photons and theself-learning features of artificial neurons to dramatically reduce thenumber of measurements required to classify sources of light. Someembodiments demonstrated the identification of light sources with onlytens of measurements at mean-photon numbers below one. This achievementrepresented a dramatic reduction in the number of photons andmeasurements of several orders of magnitude with respect to conventionalschemes for quantum state characterization. Additional embodimentsinclude smart quantum cameras, for example. These cameras can rely onthe technology described above and in the following references, whichare incorporated herein by reference. This is a novel quantum technologyand the first demonstration of a smart quantum camera can dramaticallychange current technologies for remote sensing and object tracking.

FIG. 2 is a schematic diagram of a smart quantum camera for remotesensing, according to an embodiment of the present invention. FIG. 2shows, in a), a smart single-pixel camera with photon-number resolutionenables the identification of the light sources illuminating anarbitrary object. As shown in FIG. 2 , light reflected from an objectwill be projected onto a digital micromirror device (DMD) where a seriesof binary patterns will be displayed. The light from the DMD will becollected by a single-mode fiber and then sent to aphoton-number-resolving detector that will measure its statisticalfluctuations. These underlying quantum fluctuations of photons will beidentified by artificial neural networks, thus enabling a fast imagereconstruction for specific light sources, as illustrated in b)-d). Itis noted that the projection of light onto random matrices in the DMDwill allow for artificial-intelligence-assisted compressive single-pixelcameras with photon-number resolution. These cameras will allow theformation of images as those shown in b)-d). The image in b) shows a redsection (indicated by an arrow) that illustrates a section of the objectilluminated by one coherent light source. The image in c) shows a yellowsection (indicated by an arrow) that illustrates the section of theobject simultaneously illuminated by two thermal light sources. Theimage in d) shows a white section (indicated by an arrow) thatillustrates the section of the object simultaneously illuminated by onethermal and two coherent light sources. This technology requires noveltheoretical models for the quantum fluctuations of light, the design ofoptimal convolutional neural networks, and the implementation ofefficient single-pixel cameras with photon-number-resolvingcapabilities. The smart quantum cameras according to some embodiments ofthe current invention can have functionalities as those described inFIG. 2 . The smart cameras can rely on novel models to describecharacteristic photon statistics produced by the scattering of photonsfrom multiple light sources (e.g., coherent light sources and/or thermallight sources).

An accurate description of these fundamental effects enables the designand implementation of artificial neural networks for classification anddiscrimination of light sources in realistic scenarios. Seminal researchin this direction and demonstrated the engineering of quantumfluctuations of multiphoton systems is described. The experimentaldemonstration of a new generation of artificial neural networks canenable the generalization of smart single-pixel quantum detectors to asmart multi-pixel quantum camera with photon-number resolution accordingto an embodiment of this invention.

Accordingly, an optical imaging system for forming images from aclassified type of light source according to another embodiment of thecurrent invention includes a plurality of light detectors arranged in apatterned array; and a processing system configured to communicate withthe plurality light detectors to receive signals to be processed toprovide an image from the classified type of light source. Theprocessing system is constructed to perform the method of any embodimentof the current invention for each of the plurality light detectors.

The imaging system and method will be described further in detail in thefollowing paragraphs. FIG. 3 is a schematic representation of anexperimental apparatus for using coherent beam of light and thermal beamof light, according to an embodiment of the present invention. Theapparatus includes using a continuous-wave (CW) laser beam that isdivided by a 50:50 beam splitter. The transmitted beam is focused onto arotating ground glass, which is used to generate pseudo-thermal lightwith super-Poissonian statistics. The beam emerging from the groundglass is collimated using a lens and attenuated by neutral-density (ND)filters to mean photon numbers below one. The attenuated beam is thencoupled into a single-mode fiber (SMF). The fiber directs photons to asuperconducting nanowire single-photon detector (SNSPD). Furthermore,the beam reflected by the beam splitter is used as a source of coherentlight. This beam, characterized by Poissonian statistics, is alsoattenuated, coupled into a SMF and detected by another SNSPD. TheSNSPDs' bias voltages are set to achieve high-efficiency photon countingwith less than five dark counts per second. The mean photon number ofthe coherent beam is matched to that of the pseudo-thermal beam oflight. In order to perform photon counting with our SNSPDs, we use thesurjective photon counting method. In this case, thetransistor-transistor logic (TTL) pulses produced by the SNSPDs weredetected and recorded by an oscilloscope. The data were divided in timebins of 1 ls, which corresponds to the coherence time of the CW laser.Moreover, the 20 ns recovery time of our SNSPDs ensured that we performmeasurements on a single-temporal-mode field. Voltage peaks above ˜0.5 Vwere considered as one photon event. The number of photons (voltagepeaks) in each time bin was counted to retrieve photon statistics. Theseevents were then used for training and testing the present ADALINEneuron and naive Bayes classifier.

The probability of finding n photons in coherent light is given by

${{P_{coh}(n)} = {e^{- \overset{¯}{n}}\left( \frac{{\overset{¯}{n}}^{n}}{n!} \right)}},$

where n denotes the mean photon number of the beam. Furthermore, thephoton statistics of thermal light is given by P_(th)(n)=n ^(n)(n+1)^(n+1). It is worth noting that the photon statistics of thermallight is characterized by random intensity fluctuations with a variancegreater than the mean number of photons in the mode. For coherent light,the maximum photon-number probability sits around n. For thermal light,the maximum is always at vacuum. However, when the mean photon number islow, the photon number distribution for both kinds of light becomessimilar. Consequently, it becomes extremely difficult to discriminateone source from the other source. The conventional approach todiscriminate light sources makes use of histograms generated through thecollection of millions of measurements. Unfortunately, this method isnot only time consuming, but also imposes practical limitations.

In order to dramatically reduce the number of measurements required toidentify light sources, we make use of an ADALINE neuron. ADALINE is asingle neural network model based on a linear processing element,proposed initially by Bernard Widrow, for binary classification. Ingeneral, the neural networks undergo two stages: training and test. Inthe training stage, ADALINE is capable of learning the correct outputs(named as output labels or classes) from a set of inputs, so-calledfeatures, by using a supervised learning algorithm. In the test stage,the ADALINE neuron produces the outputs of a set of inputs that were notin the training data, taking as reference the acquired experience in thetraining stage. Although we tested architectures far more complex than asingle neuron for the identification of light sources, we found that asimple ADALINE offers a perfect balance between accuracy and simplicity.The structure of the ADALINE model is shown in FIG. 1 . The neuron inputfeatures are denoted by P(n), which corresponds to the probability ofdetecting n photons, in a single measurement event, for a given lightsource, namely coherent or thermal. Furthermore, the parameters co, arethe synaptic weights and b is a bias term. In the training period, theseparameters are optimized through the learning rule by using the errorbetween the target output and neuron's output as reference. For thebinary classification (coherent or thermal), the neuron's output is fedinto the identity activation function, and subsequently to the thresholdfunction.

To train the ADALINE, we make use of the so-called delta learning rule,in combination with a database of experimentally measured photon-numberdistributions, considering different mean photon numbers: n=0.44, 0.53,0.67, 0.77. The database for each mean photon number was divided intosubsets comprising 10, 20, . . . , 150, 160 data points. The ADALINEneurons are thus prepared by using one thousand of those subsets, where70% are devoted to training and 30% to testing. In all cases, thetraining was stopped after 50 epochs.

We have established the baseline performance for our ADALINE neuron byusing naive Bayes classifier. This is a simple classifier based onBayes' theorem. Throughout this article, we assume that each measurementis independent. Moreover, we represent the measurement of the photonnumber sequence as a vector x=(x₁, . . . , x_(k)). Then, the probabilityof this sequence generated from coherent or thermal light is given by p(C_(j)|x₁, . . . , x_(k)) where C_(j) could denote either coherent orthermal light. Using Bayes' theorem, the conditional probability can bedecomposed as

${p\left( {C_{j}{❘x}} \right)} = {\frac{{p\left( c_{j} \right)}{p\left( {x{❘C_{j}}} \right)}}{p(x)}.}$

By using the chain rule for conditional probability, we have p(C_(k)|x₁,. . . , x_(k))=p(C_(j))Π_(i=1) ^(k)p(x_(i)|C_(j)). Since our lightsource is either coherent or thermal, we assume p(C_(j))=0.5. Thus, itis easy to construct a naive Bayes classifier, where one picks thehypothesis with the highest conditional probability p(C_(j)|x). We usedtheoretically generated photon-number probability distributions as theprior probability p(x_(i)|C_(j)), and used the experimental data as thetest data.

FIGS. 4A-4D shows a set of histograms displaying theoretical andexperimental photon number probability distributions for coherent andthermal light beams with different mean photon numbers, according to anembodiment of the present invention. These histograms show that ourexperimental results are in excellent agreement with theory. The photonnumber distributions illustrate the difficulty in discriminating lightsources at low-light levels even when large sets of data are available.In FIGS. 4A-4D, we compare the histograms for the theoretical andexperimental photon number distributions for different mean photonnumbers n=0.40, 0.53, 0.67 and 0.77. The bar plots are generated byexperimental data with one million measurements for each source. Thecurves in each of the panels represent the expected theoretical photonnumber distributions for the corresponding mean photon numbers. FIGS.4A-4D show excellent agreement between theory and experiment whichdemonstrates the accuracy of our surjective photon counting method.Furthermore, we can also observe the effect of the mean photon number onthe photon number probability distributions. As shown in FIG. 4A, it isevident that millions of measurements enable one to discriminate lightsources. On the other hand, FIG. 4D shows a situation in which thesource mean-photon number is low. In this case, the discrimination oflight sources becomes cumbersome, even with millions of measurements. Inorder to illustrate the difficulty of using limited sets of data todiscriminate light sources at low mean photon numbers, we restrict thesize of our dataset to 10, 20, 50, 100 and 100000.

FIGS. 5A-5B show the probability distributions of coherent and thermallight, for varying dataset sizes (10, 20, 50, 100, 10 k), according toan embodiment of the present invention. Data used here is randomlyselected from of the measurement presented in FIG. 4A. As shown in FIGS.5A-5B, the photon number distributions obtained with limited number ofmeasurements do not resemble those in the histograms shown in FIG. 4A,for both coherent and thermal light beams.

FIG. 6 is a plot of the overall accuracy of light discrimination versusthe number of data points used in naive Bayes classifier. In FIG. 6 ,the curves represent the accuracy of light discrimination for n=0.40(red line), n=0.53 (blue line), n=0.67 (green line) and n=0.77 (orangeline). The error bars are generated by dividing the data into tensubsets. For example, when n=0.40, the accuracy of discriminationincreases from approximately 61% to 90% as we increase the number ofdata points from 10 to 160. It is worth noting that even with smallincrease in number of measurements, the naive Bayes classifier starts tocapture the characteristic feature of different light sources, given bydistinct sequences of photon number events. This can be understood aslarger sets of data contain more information pertaining to theprobability distribution. Furthermore, mean photon number of the lightfield significantly changes the discrimination accuracy profile. As themean photon number increases, the overall accuracy converges fastertowards 100% as expected. This is due to the fact that the photon numberprobability distributions become more distinct at higher mean photonnumber.

FIG. 7 is a plot of the overall accuracy of light discrimination versusthe number of data points used in ADALINE, according to an embodiment ofthe present invention. The curves represent the accuracy of lightdiscrimination for n=0.40 (red line), n=0.53 (blue line), n=0.67 (greenline) and n=0.77 (orange line). The error bars represent the standarddeviation of the training stages. Using only 10 data points, ADALINEleads to an average accuracy between 61%-65% for n=0.40; whereas for 160data points, the accuracy is greater than 90%. The comparison of FIG. 6and FIG. 7 reveals that ADALINE and naive Bayes classifier exhibitsimilar accuracy levels. However, ADALINE requires far lesscomputational resources than naive Bayes classifier. As one mightexpect, in both cases, the accuracy increases with the number of datapoints and mean photon numbers. Interestingly, the convergence rate fornaive Bayes is slightly higher than that of ADALINE classifier. For lowmean photon numbers, such as n=0.40, the improvement in accuracy scaleslinearly for naive Bayes classifier, as opposed to almost logisticgrowth that has our ADALINE. This implies that at low mean photonnumbers ADALINEs outperform naive Bayes classifier in the sense that theADALINE uses much less computational resources than the Bayesclassifier.

FIGS. 8A-8D show a 3-dimensional (3D) projection of the feature space onthe plane (P(0), P(1), P(2)) for different mean photon numbers,according to an embodiment of the present invention. FIG. 8A shows a 3Dprojection for a number of photons (a) n=0.4. FIG. 8B shows a 3Dprojection for a number of photons (b) n=0.53. FIG. 8C shows a 3Dprojection for a number of photons (c) n=0.67. FIG. 8D shows a 3Dprojection for a number of photons (d) n=0.77. The blue pointscorrespond to photon statistics of coherent light, whereas the red starsdescribe photon statistics of thermal light. In all cases the number ofdata points is fixed at M=60.

To understand why a single ADALINE neuron is enough for lightdiscrimination, we first realize that ADALINE is a linear classifier.Therefore, the decision surface is expressed by a seven-dimensionalhyper-plane, defined by the seven P(n) (with n=0, 1, . . . , 6)features. Interestingly, one can find that the datasets at the space ofprobability-distribution values are linearly separable. This can be seenfrom FIGS. 8A-8D, where we plot the projection of the feature space on athree-dimensional sub-space defined by (P(0), P(1), P(2)) consideringdifferent mean photon numbers n=0.4, 0.53, 0.67 and 0.77 (the number ofdata points is fixed at M=60 in all cases). Within this subspace, thedatasets corresponding to the photon statistics of thermal (red stars)and coherent (blue points) lights separate each other as n increases.This effect is more evident when the number of data points is increased,and the mean photon number remains fixed at n=0.77 (see, FIGS. 9A-9D).Evidently, the fact that both, thermal and coherent light form two welllinearly separated classes makes ADALINE the optimum classifier forlight identification.

FIGS. 9A-9D shows 3D Projection of the feature space on the plane (P(0),P(1), P(2)) for different number of data points, according to anembodiment of the present invention. FIG. 9A shows a 3D projection using10 data points. FIG. 9B shows a 3D projection using 60 data points. FIG.9C shows a 3D projection using 160 data points. FIG. 9D shows a 3Dprojection using 600 data points. The blue points correspond to photonstatistics of coherent light, whereas the red stars describe photonstatistics of thermal light. In all cases, the mean photon number is setto n=0.77.

In embodiments of the present invention, we evaluate two additionalmachine-learning (ML) algorithms, namely a one-dimensional convolutionalneural network (1D CNN) and a multilayer neural network (MNN). Despiteboth algorithms are effective to identify light sources, they areanalytically and computationally more sophisticated than the simpleADALINE model, but their recognition rates do not present substantialdifferences.

FIG. 10A is a schematic structure representation one-dimensionalconvolutional neural network 1D-CNN), according to an embodiment of thepresent invention. FIG. 10B is a schematic structure representation of amultilayer neural network (MNN) used for demonstration of light sourceidentification, according to another embodiment of the presentinvention. A convolutional neural network (CNN) is a deep learningalgorithm that extracts automatically relevant features of the input.The present one-dimensional convolutional neural network (1D-CNN) iscomposed by two 1D-convolutional layers that extract the low andhigh-level features of the input. Outcomes from these two layers aresubsequently fed into a convolutional layer sandwiched between twomax-pooling layers. The pooling layers downsample the inputrepresentation, and therefore its dimensionality, leading to acomputational simplification by removing redundant and unnecessaryinformation. The activation function, implemented in all layers, is therectified linear unit function (ReLU). Finally, a fully connected and aflattening layer precedes the output layer consisting of two softmaxfunctions, whose outputs are the probability distributions over labels.

On the other hand, the multilayer neural network (MNN) belongs to aclassical machine learning algorithm, where the feature vector ismanually determined. In the present case, this vector is given by theprobabilities of the photon number distribution, P(n). As depicted inFIG. 10B, the model corresponds to a two-layer feed-forward network: thehidden layer contains ten sigmoid neurons and the output layer consistsof a softmax function. To determine a suitable neuron number in thehidden layer of the MNN, we trained different MNNs by changing theneuron number in the hidden layer and followed the accuracy values foreach net.

FIGS. 11A-11B are plots of the overall accuracy of light discriminationversus the number of neurons in the hidden layer of the MNN byconsidering two different mean photon numbers, according to anembodiment of the present invention. FIG. 11A is a plot of the accuracyof light discrimination versus number of neurons for a mean photonnumber n=0.4. FIG. 11B is a plot of the accuracy of light discriminationversus number of neurons for a mean photon number n=0.77. The error barsrepresent the standard deviation of the training stages. Note that inboth cases, the accuracy becomes lower as the number of neuronsincreases. This is because many neurons lead to over-parameterization,causing poor generalization of the test-stage data. Additionally, as thenumber of neurons increases, the training becomes computationally moreintensive. All the MNNs were trained by using the scaled conjugategradient backpropagation method where the cross-entropy was employed asthe cost function. Since the output of sigmoid neurons is ranged in theinterval [0,1], the cross-entropy function is ideal for theclassification task. The network training was stopped after 200 epochs.

FIGS. 12A-12B are plots of the overall accuracy of light discriminationversus the number of data points, according an embodiment of the presentinvention. FIG. 12A is a plot of the accuracy versus the number ofpoints used in (a) 1D-CNN. FIG. 12B is a plot of the accuracy versus thenumber of points used in (b) MNN. The curves represent the accuracy oflight discrimination for n=0.40 (red line), n=0.53 (blue line), n=0.67(green line) and n=0.77 (orange line). The error bars represent thestandard deviation of the training epochs for 1D-CNN and training stagesfor MNN.

Another aspect of the present invention is to improve or enhance theresolution of optical imaging systems. The spatial resolution of opticalimaging systems is established by the diffraction of photons and thenoise associated with their quantum fluctuations. For over a century,the Abbe-Rayleigh criterion has been used to assess thediffraction-limited resolution of optical instruments. At a morefundamental level, the ultimate resolution of optical instruments isestablished by the laws of quantum physics through the Heisenberguncertainty principle. In classical optics, the Abbe-Rayleigh resolutioncriterion stipulates that an imaging system cannot resolve spatialfeatures smaller than λ/2NA. In this case, X represents the wavelengthof the illumination field, and NA describes numerical aperture of theoptical instrument. Given the implications that overcoming theAbbe-Rayleigh resolution limit has for multiple applications, such as,microscopy, remote sensing, and astronomy, there has been an enormousinterest in improving the spatial resolution of optical systems.Recently, optical super-resolution has been demonstrated throughdecomposition of spatial eigenmodes.

For almost a century, the importance of phase over amplitude informationhas constituted established knowledge for optical engineers. Recently,this idea has been extensively investigated in the context of quantummetrology. More specifically, it has been demonstrated that phaseinformation can be used to surpass the Abbe-Rayleigh resolution limitfor the spatial identification of light sources. For example, phaseinformation can be obtained through mode decomposition by usingprojective measurements or demultiplexing of spatial modes. Naturally,these approaches require a priori information regarding the coherenceproperties of the, in principle, “unknown” light sources. Furthermore,these techniques impose stringent requirements on the alignment andcentering conditions of imaging systems. Despite these limitations,most, if not all, the current experimental protocols have relied onspatial projections and demultiplexing in the Hermite-Gaussian,Laguerre-Gaussian, and parity basis.

The quantum statistical fluctuations of photons establish the nature oflight sources. As such, these fundamental properties are not affected bythe spatial resolution of an optical instrument. Here, we demonstratethat measurements of the quantum statistical properties of a light fieldenable imaging beyond the Abbe-Rayleigh resolution limit. This isperformed by exploiting the self-learning features of artificialintelligence to identify the statistical fluctuations of photonmixtures. More specifically, we demonstrate a smart quantum camera withthe capability to identify photon statistics at each pixel. For thispurpose, we introduce a universal quantum model that describes thephoton statistics produced by the scattering of an arbitrary number oflight sources. This model is used to design and train artificial neuralnetworks for the identification of light sources. Remarkably, our schemeenables us to overcome inherent limitations of existing super-resolutionprotocols based on spatial mode projections and multiplexing.

FIGS. 13A and 13B is a conceptual illustration and schematic of anotherexperimental setup to demonstrate super-resolving imaging, according toan embodiment of the present invention. The illustration in FIG. 13Adepicts a scenario where diffraction limits the resolution of an opticalinstrument for remote imaging. The present scheme is capable ofidentifying the corresponding photon fluctuations and theircombinations, for example coherent-thermal (CT1, CT2), thermal-thermal(TT) and coherent-thermal-thermal (CTT). This capability allows to boostthe spatial resolution of optical instruments beyond the Abbe-Rayleighresolution limit. The experimental setup shown in FIG. 13B is designedto generate two independent thermal source and one coherent lightsource. The three sources are produced from a continuous-wave (CW) laserat 633 nm. The CW laser beam is divided by two beam splitters (BS) togenerate three spatial modes, two of which are then passed throughrotating ground glass (RGG) disks to produce two independent thermallight beams. The three light sources, with different photon statistics,are attenuated using neutral density (ND) filters and then combined tomimic a remote object such as the one shown in the inset of FIG. 13B.This setup enables the generation of multiple sources with tunablestatistical properties. The generated target beam is then imaged onto adigital micro-mirror device (DMD) that can be used to perform rasterscanning. The photons reflected off the DMD are collected and measuredby a single-photon detector. The present protocol is formalized byperforming photon-number-resolving detection.

The schematic behind the experiment is depicted in FIGS. 13A and 13B.This camera utilizes an artificial neural network to identify the photonstatistics of each point source that constitutes a target object. Thedescription of the photon statistics produced by the scattering of anarbitrary number of light sources is achieved through a general modelthat relies on the quantum theory of optical coherence. We use thismodel to design and train a neural network capable of identifying lightsources at each pixel of our camera. This is achieved by performingphoto-number-resolving detection. The sensitivity of the present camerais limited by the photon fluctuations of the detected field.

In general, realistic imaging instruments deal with the detection ofmultiple light sources. These sources can be either distinguishable orindistinguishable. The combination of indistinguishable sources can berepresented by either coherent or incoherent superpositions of lightsources characterized by Poissonian (coherent) or super-Poissonian(thermal) statistics. In our model, we first consider theindistinguishable detection of N coherent and M thermal sources. Forthis purpose, we make use of the P-function P_(coh)(γ)=δ²(γ−α_(k)) tomodel the contributions from the kth coherent source with thecorresponding complex amplitude α_(k). The total complex amplitudeassociated to the superposition of an arbitrary number of light sourcesis given by α_(tot)=Σ_(k=1) ^(N)α_(k). In addition the P-function forthe lth thermal source, with the corresponding mean photon numbers m_(l), is defined as P_(th)(γ)=(πm _(l))⁻¹ exp(−|γ|²/m _(l)). The totalnumber of photons attributed to the M number of thermal sources isdefined as m_(tot)=Σ_(l=1) ^(M) m _(l). These quantities allow us tocalculate the P-function for the multisource system as equation (3).

$\begin{matrix}{{P_{{th} - {coh}}(\gamma)} = {\int{\ldots{\int{{P_{N + M}\left( {\gamma - \gamma_{N + M - 1}} \right)} \times \left\lbrack {\prod\limits_{i = 2}^{N + M - 1}{{P_{i}\left( {\gamma_{i} - \gamma_{i - 1}} \right)}d^{2}\gamma_{i}}} \right\rbrack{P_{1}\left( \gamma_{1} \right)}d^{2}\gamma_{1}}}}}} & (3)\end{matrix}$

This approach enables the analytical description of the photon-numberdistribution p_(th−coh)(n) associated to the detection of an arbitrarynumber of indistinguishable light sources. This is calculated asp_(th−coh)(n)=

n|{circumflex over (ρ)}_(th−coh)|n

, where p_(th−coh)=∫P_(th−coh)(γ)|γ

γ|d²γ. After algebraic manipulation, we obtain the followingphoton-number distribution (4).

$\begin{matrix}{{p_{{th} - {coh}}(n)} = {\frac{\left( m_{tot} \right)^{n}\exp\left( {- \frac{\left( {❘\alpha_{tot}❘} \right)^{2}}{m_{tot}}} \right)}{{\pi\left( {m_{tot} + 1} \right)}^{n + 1}} \times {\sum\limits_{k = 0}^{n}{\frac{1}{k{!{\left( {n - k} \right)!}}}{\Gamma\left( {\frac{1}{2} + n - k} \right)}{\Gamma\left( {\frac{1}{2} + k} \right)} \times {\,_{1}F_{1}}\left( {{\frac{1}{2} + n - k};\frac{1}{2};\frac{\left( {{Re}\left\lbrack \alpha_{tot} \right\rbrack} \right)^{2}}{m_{tot}\left( {m_{tot} + 1} \right)}} \right){\,_{1}F_{1}}\left( {{\frac{1}{2} + k};\frac{1}{2};\frac{\left( {{Im}\left\lbrack \alpha_{tot} \right\rbrack} \right)^{2}}{m_{tot}\left( {m_{tot} + 1} \right)}} \right)}}}} & (4)\end{matrix}$

where Γ(z) and ₁F₁(a; b; z) are the Euler gamma and the Kummer confluenthypergeometric functions, respectively. This probability functionenables the general description of the photon statistics produced by anyindistinguishable combination of light sources. Thus, the photondistribution produced by the distinguishable detection of N lightsources can be simply obtained by performing a discrete convolution ofequation (4) as following equation (5).

$\begin{matrix}{{p_{tot}(n)} = {\sum\limits_{m_{1} = 0}^{n}{\sum\limits_{m_{2} = 0}^{n - m_{1}}{\ldots{\sum\limits_{m_{N - 1} = 0}^{m - {\sum_{j = 1}^{N - 1}m_{j}}}{{p_{1}\left( m_{1} \right)}{p_{2}\left( m_{2} \right)}\ldots{p_{N - 1}\left( m_{N - 1} \right)}{{p_{N}\left( {n - {\sum\limits_{j = 1}^{n - 1}m_{j}}} \right)}.}}}}}}} & (5)\end{matrix}$

The combination of equation (4) and equation (5) allows theclassification of photon-number distributions for any combination oflight sources.

FIG. 14A shows a scheme of the two-layer neural network used to identifythe photon statistics produced by a combination of three sources,according to an embodiment of the present invention. The computationalmodel consists of an input layer, a hidden layer of sigmoid neurons, anda Softmax output layer. The training of our neural network throughequation (4) and equation (5) enables the efficient identification offive classes of photon statistics. Each class is characterized by a g⁽²⁾function, which is defined by a specific combination of light sources.In the present experiment, these classes correspond to thecharacteristic photon statistics produced by coherent or thermal lightsources and their combinations. For example, coherent-thermal,thermal-thermal, or coherent-thermal-thermal.

FIG. 14B shows the performance of the present neural network as afunction of the number of data samples used each time in the testingprocess, according to an embodiment of the present invention. Theclassification accuracy for the five possible complex classes of lightis 80% with 100 data points. Remarkably, the performance of the neuralnetwork increases to approximately 95% when we use 3500 data points ineach test sample.

We demonstrate a proof-of-principle quantum camera using theexperimental setup shown in FIG. 13B. For this purpose, we use acontinuous-wave laser at 633 nm to produce either coherent, orincoherent superpositions of distinguishable, indistinguishable, orpartially distinguishable light sources. In this case, the combinationof photon sources acts as our target object. Then, we image our targetobject onto a digital micro-mirror device (DMD) that is used toimplement raster scanning. This is implemented by selectively turning onand off groups of pixels in our DMD. The light reflected of the DMD ismeasured by a single-proton detector that allows us to performphoton-number-resolving detection.

The equations above allow us to implement a multilayer feed-forwardnetwork for the identification of the quantum photon fluctuations of thepoint sources of a target object. As shown in FIG. 14A, the structure ofthe network includes a group of interconnected neurons arranged inlayers. In this case, the input features represent the probabilities ofdetecting n photons at a specific pixel, p(n), whereas the neurons inthe last layer correspond to the classes to be identified. The inputvector is then defined by twenty-one features corresponding to n=0,1, .. ., 20. We define five classes, which can be directly described throughequation (4) and equation (5) if the brightness of our sources remainsconstant. However, if the brightness is modified, the classes can bedefined through the g⁽²⁾=1+(

(Δ{circumflex over (n)})²

−

{circumflex over (n)}

)/

{circumflex over (n)}

², which is intensity-independent. The parameters in the g(2) functioncan also be calculated from equations (4) and equation (5). It isimportant to mention that the output neurons provide a probabilitydistribution over the predicted classes .

We test the performance of the present neural network through theclassification of a complex mixture of photons produced by thecombination of one coherent with two thermal light sources. The accuracyof our trained neural network is reported in FIG. 14B. In our setup, thethree partially overlapping sources form five classes of light withdifferent mean photon numbers and photon statistics. We exploit thefunctionality of our artificial neural network to identify theunderlying quantum fluctuations that characterize each kind of light. Wecalculate the accuracy as the ratio of true positive and true negativeto the total of input samples during the testing phase. FIG. 14B showsthe overall accuracy as a function of the number of data points used tobuild the probability distributions for the identification of themultiple light sources using a supervised neural network. Theclassification accuracy for the mixture of three light sources is 80%with 100 photon-number-resolving measurements. The performance of theneural networks increases to approximately 95% when we use 3500 datapoints to generate probability distributions.

FIGS. 15A-15F show various aspects of the experimental super-resolvingimaging, according to an embodiment of the present invention. Thecontour plot in FIG. 15A shows the combined intensity profile of thethree partially distinguishable sources. As stipulated by the

Abbe-Rayleigh resolution criterion, the transverse separations among thesources forbid their identification. The contour plot shown in FIG. 15B,shows that the present smart quantum camera enables super-resolvingimaging of the remote sources. In FIGS. 15C and 15D, we show anotherexperimental realization of our protocol for a different distribution oflight sources. In this case, two small sources are located inside thepoint-spread function of a third light source. FIGS. 1E and 15Fcorrespond to the inferred spatial distributions based on theexperimental pixel-by-pixel imaging used to produce FIG. 15B and FIG.15D. The insets in FIG. 15E and FIG. 15F show photon-number probabilitydistributions for three pixels, the theory bars were obtained throughequation (4) and equation (5). These results demonstrate the potentialof our technique to outperform conventional diffraction-limited imaging.

As demonstrated in FIGS. 15A-15F, the identification of the quantumphoton fluctuations at each pixel of our camera enables us todemonstrate super-resolving imaging. In our experiment we prepared eachsource to have a mean photon number between 1 and 1.5 for the brightestpixel. The raster-scan image of a target object composed of multiplepartially distinguishable sources in FIG. 15A illustrates theperformance of conventional imaging protocols limited by diffraction. Inthis case, it is practically impossible to identify the multiple sourcesthat constitute the target object. Remarkably, as shown in FIG. 15B, ourprotocol provides a dramatic improvement of the spatial resolution ofthe imaging system. In this case, it becomes clear the presence of thethree emitters that form the remote object. The estimation ofseparations among light sources is estimated through a fit over theclassified pixel-by-pixel image. In FIG. 15C and FIG. 15D, wedemonstrate the robustness of our protocol by performing super-resolvingimaging for a different configuration of light sources. In this case,two small sources are located inside the point-spread function of athird light source. As shown in FIG. 15C, the Abbe-Rayleigh limitforbids the identification of light sources. However, we demonstratesubstantial improvement of spatial resolution in FIG. 15D. The plots inFIG. 15E and FIG. 15F correspond to the inferred spatial distributionsbased on the experimental pixel-by-pixel imaging used to produce FIG.15B and FIG. 15D. The insets in FIG. 15E and FIG. 15F show photon-numberprobability distributions for three pixels. The theoreticalphoton-number distributions in FIG. 15E and FIG. 15F are obtainedthrough a procedure of least square regression. Our scheme enables theuse of the photon-number distributions or their corresponding g⁽²⁾ tocharacterize light sources. This allows us to determine each pixel'scorresponding statistics, regardless of the mean photon numbers of thesources in the detected field.

FIG. 16 shows a comparison between the spatial resolution of our cameraand direct imaging, according to an embodiment of the present invention.The distance is normalized by the beam radius for easy identification ofthe Abbe-Rayleigh limit. The red line is the result of a Monte-Carlosimulation for traditional intensity based direct imaging. The plateauis the area where the algorithm becomes unstable and fits to oneGaussian. The dotted blue line represents the limit for oursuper-resolving imaging method, where perfect classification of eachpixel is assumed. The blue dots represent the experimental datacollected with our camera for super-resolving imaging. The experimentalpoints demonstrate the potential of our technique for identifyingspatial features beyond the Abbe-Rayleigh resolution criterion.

We now provide a quantitative characterization of our super-resolvingimaging scheme based on the identification of photon statistics. Wedemonstrate that our smart camera for super-resolving imaging cancapture small spatial features that surpass the resolution capabilitiesof conventional schemes for direct imaging. Consequently, as shown inFIG. 16 , the present camera enables the possibility of performingimaging beyond the Abbe-Rayleigh criterion. In this case, we performedmultiple experiments in which a superposition of partiallydistinguishable sources were imaged. The superposition was preparedusing one coherent and one thermal light source. In FIG. 16 , we plotthe predicted transverse separation s normalized by the Gaussian beamwaist radius w₀ for both protocols. Here w₀=λ/πNA, this parameter isdirectly obtained from our experiment. As shown in FIG. 16 , ourprotocol enables one to resolve small spatial separations between thesources even for diffraction-limited conditions. As expected for largerseparation distances, the performance of our protocol matches theaccuracy of intensity measurements. For completeness, we also performedMonte Carlo simulations of our experiment, which show an excellentagreement with our experimental data.

Derivation of the Many-source Photon Statistics: Let us start byconsidering the indistinguishable detection of N coherent and M thermalindependent sources. To obtain the combined photon distribution, we makeuse of the Glauber-Sudarshan theory of coherence. Thus, we start bywriting the P-functions associated to the fields produced by theindistinguishable coherent and thermal sources, that is, we writefollowing equations (6) and (7).

P _(coh)(α)=∫P _(N) ^(coh)(α−α_(N−1))P _(N−1) ^(coh)(α_(N−1)−α_(N−2)) .. . P ₂ ^(coh)(α₂−α₁)P ₁ ^(coh)(α₁)d ²α_(N−1) d ²α_(N−2) . . . d ²α₂ d²α₁,   (6)

P _(th)(α)=∫P _(M) ^(th)(α−α_(M−1))P _(M−1) ^(th)(α_(M−1)−α_(M−2)) . . .P ₂ ^(th)(α₂−α₁)P ₁ ^(th)(α₁)d ²α_(M−1) d ²α_(M−2) . . . d ²α₂ d ²α₁,  (7)

with P_(coh)(α) and P_(th)(α) standing for the P-functions of thecombined N-coherent and M-thermal sources, respectively. In bothequations, a stands for the complex amplitude as defined for coherentstates |α

, and the individual-source P-functions are defined as followingequations (8) and (9).

$\begin{matrix}{{{P_{k}^{coh}(\alpha)} = {\delta^{2}\left( {\alpha - \alpha_{k}} \right)}},} & (8)\end{matrix}$ $\begin{matrix}{{{P_{l}^{th}(\alpha)} = {\frac{1}{\pi{\overset{¯}{m}}_{l}}\exp\left( {{- {❘\alpha ❘}^{2}}/{\overset{¯}{m}}_{l}} \right)}},} & (9)\end{matrix}$

where P_(k) ^(coh)(α) corresponds to the P-function of kth coherentsource, with mean photon number n _(k)=|α_(k)|², and P_(l) ^(th)(α)describes the lth thermal source, with mean photon number m _(l). Now,by substituting equation (8) into equation (6), and equation (9) intoequation (7), we obtain equation (10) and equation (11), as follows.

$\begin{matrix}{{{P_{coh}(\alpha)} = {\delta^{2}\left( {\alpha - {\sum\limits_{k = 1}^{N}\alpha_{k}}} \right)}},} & (10)\end{matrix}$ $\begin{matrix}{{{P_{th}(\alpha)} = {\left( \frac{1}{\pi{\sum}_{l = 1}^{M}{\overset{¯}{m}}_{l}} \right)\exp\left( {- \frac{{❘\alpha ❘}^{2}}{{\sum}_{l = 1}^{M}{\overset{¯}{m}}_{l}}} \right)}},} & (11)\end{matrix}$

We can finally combine the thermal and coherent sources by writingequation (12), as follows.

P _(th−coh)(α)=∫P _(th)(α−α′)P _(coh)(α′)d ²α′  (12)

Note that this expression enables the analytical description for thephoton-number distribution of an arbitrary number of indistinguishablesources measured by a quantum detector. More specifically, we can writeequation (13), as follows.

P _(th−coh)(n)=

n|{circumflex over (ρ)} _(th−coh) |n

,   (13)

where

{circumflex over (ρ)}_(th−coh) =∫P _(th−coh)(α)|α

α|d ²α,   (14)

describes the density matrix of the quantum states of the combinedthermal-coherent field at the quantum detector. Thus, by substitutingequation (12) into equation (14) and equation (13), we find that thephoton distribution of the combined fields is given by equation (15), asfollows.

$\begin{matrix}{{p_{{th} - {coh}}(n)} = {\frac{\left( m_{tot} \right)^{n}\exp\left( {- \frac{\left( {❘\alpha_{tot}❘} \right)^{2}}{m_{tot}}} \right)}{{\pi\left( {m_{tot} + 1} \right)}^{n + 1}} \times {\sum\limits_{k = 0}^{n}{\frac{1}{k{!{\left( {n - k} \right)!}}}{\Gamma\left( {\frac{1}{2} + n - k} \right)}{\Gamma\left( {\frac{1}{2} + k} \right)} \times {\,_{1}F_{1}}\left( {{\frac{1}{2} + n - k};\frac{1}{2};\frac{\left( {{Re}\left\lbrack \alpha_{tot} \right\rbrack} \right)^{2}}{m_{tot}\left( {m_{tot} + 1} \right)}} \right){\,_{1}F_{1}}\left( {{\frac{1}{2} + k};\frac{1}{2};\frac{\left( {{Im}\left\lbrack \alpha_{tot} \right\rbrack} \right)^{2}}{m_{tot}\left( {m_{tot} + 1} \right)}} \right)}}}} & (15)\end{matrix}$

with m_(tot)=Σ_(l=1) ^(M) m _(l) and α_(tot)=Σ_(k=1) ^(N)α_(k). In thisfinal result, Γ(z) and ₁F₁(a; b; z) are the Euler gamma and the Kummerconfluent hypergeometric functions, respectively.

Training of Neural Networks: For the sake of simplicity, we split thefunctionality of our neural network into two phases: the training andtesting phase. In the first phase, the training data is fed to thenetwork multiple times to optimize the synaptic weights through a scaledconjugate gradient back-propagation algorithm. This optimization seeksto minimize the Kullback-Leibler divergence distance between predictedand the real target classes. At this point, the training is stopped ifthe loss function does not decrease within 1000 epochs. In the testphase, we assess the performance of the algorithm by introducing anunknown set of data during the training process. For both phases, weprepare a data-set consisting of one thousand experimental measurementsof photon statistics for each of the five classes. This process isformalized by considering different numbers of data points: 100, 500, .. . , 9500, 10000. Following a standardized ratio for statisticallearning, we divide our data into training (70%), validation (15%), andtesting (15%) sets. The networks were trained using the neural networktoolbox in MATLAB, which runs on a computer Intel Core i7-4710MQ CPU(@2.50 GHz) with 32 GB of RAM.

Fittings: To determine the optimal fits for FIG. 15E and FIG. 15F, wedesign a search space based on equations (4) and (5). To do so we firstfound the mean photon number of the input pixel, which will later beapplied to constrain the search space. From here we allowed for theexistence of up to three distinguishable modes which will be combinedaccording to equation (5). Each of the modes contains anindistinguishable combination of up to one coherent and two thermalsources whose number distribution is given by equation (4). The totalcombination results in partially distinguishable combination andprovides the theoretical model for our experiment. From here our searchspace is

√{square root over (Σ_(n=0)(p_(exp)(n)−p_(th)(n|{right arrow over(n)}_(1,t),{right arrow over (n)}_(2,t),{right arrow over (n)}_(c)))²)},where {right arrow over (n)}_(i,t) and {right arrow over (n)}_(c) arethe mean photon numbers of that each thermal or coherent sourcecontributes to each distinguishable mode respectively. The mean photonnumbers of each source must add up to the experimental mean photonnumber, constraining the search. A linear search was then performed overthe predicted mean photon numbers and the minimum was returned,providing the optimal fit.

Monte-Carlo Simulation of the Experiment: To demonstrate a consistentimprovement over traditional methods, we also simulated the experimentusing two beams, a thermal and a coherent, with Gaussian point spreadfunctions over a 128×128 grid of pixels. At each pixel, the mean photonnumber for each source is provided by the Gaussian point spreadfunction, which is then used to create the appropriate distinguishableprobability distribution as given in equation (5), creating a 128×128grid of photon number distributions. The associated class data for thesedistributions will then be fitted to a set of pre-labeled disks using agenetic algorithm. This recreates our method in the limits of perfectclassification. Each of these distributions is then used to simulatephoton-number resolving detection. This data is then used to create anormalized intensity for the classical fit. We fit the image to acombination of Gaussian PSFs. This process is repeated ten times foreach separation in order to average out fluctuations in the fitting.When combining the results of the intensity fits they are first dividedinto two sets. One set has the majority of fits returns a singleGaussian, while the other returned two Gaussian the majority of thetime. The set identified as only containing a single Gaussian is thenset at the Abbe-Rayleigh diffraction limit, while the remaining data isused in a linear fit. This causes the sharp transition between the twosets of data.

We demonstrated a robust quantum camera that enables super-resolvingimaging beyond the Abbe-Rayleigh resolution limit. The demonstratedprotocol exploits the self-learning features of artificial intelligenceto identify the statistical fluctuations of truly unknown mixtures oflight sources. Our smart camera relies on a general model based on thetheory of quantum coherence to describe the photon statistics producedby the scattering of an arbitrary number of light sources. Wedemonstrated that the measurement of the quantum statisticalfluctuations of photons enables us to overcome inherent limitations ofexisting super-resolution protocols based on spatial mode projections.We believe that our work represents a new paradigm in the field ofoptical imaging with important implications for microscopy, remotesensing, and astronomy.

For more than twenty years, there has been an enormous interest inreducing the number of photons and measurements required to performimaging, remote sensing and metrology at extremely low-light levels. Inthis regard, photonic technologies operating at low-photon levelsutilize weak photon signals that make them vulnerable against detectionof environmental photons emitted from natural sources of light. Indeed,this limitation has made unfeasible the realistic implementation of thisfamily of technologies. So far, this vulnerability has been tackledthrough conventional approaches that rely on the measurement ofcoherence functions, the implementation of thresholding and quantumstate tomography. Unfortunately, these approaches to characterizephoton-fluctuations rely on the acquisition of large number ofmeasurements that impose constraints on the identification of lightsources. Here, for the first time, we have demonstrated a smart protocolfor discrimination of light sources at mean photon numbers below one.Embodiments of the present invention demonstrate a dramatic improvementof several orders of magnitude in both the number of photons andmeasurements required to identify light sources. Furthermore, ourresults indicate that a single artificial neuron outperforms naive Bayesclassifier at low-light levels. Interestingly, this neuron has simpleanalytical and computational properties that enable low-complexity andlow-cost implementations of our technique. The present method and systemhas important implications for multiple photonic technologies, such asLIDAR and microscopy of biological materials.

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While various embodiments of the present invention have been describedabove, it should be understood that they have been presented by way ofexample only, and not limitation. Thus, the breadth and scope of thepresent invention should not be limited by any of the above-describedillustrative embodiments, but should instead be defined only inaccordance with the following claims and their equivalents.

The embodiments illustrated and discussed in this specification areintended only to teach those skilled in the art how to make and use theinvention. In describing embodiments of the disclosure, specificterminology is employed for the sake of clarity. However, the disclosureis not intended to be limited to the specific terminology so selected.The above-described embodiments of the disclosure may be modified orvaried, without departing from the invention, as appreciated by thoseskilled in the art in light of the above teachings. It is therefore tobe understood that, within the scope of the claims and theirequivalents, the invention may be practiced otherwise than asspecifically described. For example, it is to be understood that thepresent disclosure contemplates that, to the extent possible, one ormore features of any embodiment can be combined with one or morefeatures of any other embodiment.

1. A method for identification of light source types, comprising:detecting individual photons for a measurement time period to provide atimes series of individual photon events; segmenting said time seriesinto a plurality of time bins; determining a number of detected photonswithin each time bin of said plurality of time bins to provide a timeseries of photon counts per time bin; determining a probabilitydistribution P(n) from said time series of photon counts per time bin,said probability distribution providing a probability of detection of nphotons, wherein n=0, 1, 2, . . . , n_(max); inputting each of values ofP(n) as a n_(max)+1 component of a feature vector into a single neuronneural network, said single neuron neural network having been previouslytrained on a plurality of light source types; and receiving as output aclassifier that has a value that identifies spa light source type,wherein an average number of photons in said plurality of time bins isless than one photon.
 2. The method according to claim 1, wherein saidlight source type is one of a coherent light source or a thermal lightsource.
 3. The method according to claim 2, wherein n_(max)=6 and saidfeature vector is a seven-component feature vector.
 4. The methodaccording to claim 3, wherein said single neuron neural networkcomprises an identity activation function and a binary classificationgiven by a threshold function to indicate a class labeled as coherent ona first side of a threshold or a class labeled thermal on a second sideof said threshold.
 5. The method according to claim 1, wherein saidplurality of time bins is less than
 100. 6. The method according toclaim 1, wherein said plurality of time bins is less than
 20. 7. Themethod according to claim 2, wherein said plurality of time bins eachhave substantially equal temporal widths and have a value selected tocorrespond to a coherence time of said coherent light source.
 8. Themethod according to claim 1, further comprising training said singleneuron neural network prior to said identifying said light source type.9. A light detection system for detecting light from a classified typeof light source, comprising: a light detector; and a processing systemconfigured to communicate with said light detector to receive signals tobe processed, wherein said light detector is configured to detectindividual photons for a measurement time period to provide a timesseries of individual photon events, and wherein said processing systemis configured to: segment said time series into a plurality of timebins; determine a number of detected photons within each time bin ofsaid plurality of time bins to provide a time series of photon countsper time bin; determine a probability distribution P(n) from said timeseries of photon counts per time bin, said probability distributionproviding a probability of detection of n photons, wherein n=0, 1, 2, .. . , n_(max); input each of values of P(n) as a n_(max)+1 component ofa feature vector into a single neuron neural network, said single neuronneural network having been previously trained on a plurality of lightsource types; and provide as output a classifier that has a value thatidentifies a light source type, wherein an average number of photons insaid plurality of time bins is less than one photon.
 10. An opticalimaging system for forming images from a classified type of lightsource, comprising: a plurality of light detectors arranged in apatterned array; and a processing system configured to communicate withsaid plurality light detectors to receive signals to be processed toprovide an image from said classified type of light source, wherein eachof said plurality of light detectors is configured to detect individualphotons for a measurement time period to provide a corresponding timesseries of individual photon events, and wherein said processing systemis configured, for each of said plurality of light detectors, to:segment each said time series into a plurality of time bins; determine anumber of detected photons within each time bin of said plurality oftime bins to provide a corresponding time series of photon counts pertime bin; determine a probability distribution P(n) from each said timeseries of photon counts per time bin, said probability distributionproviding a probability of detection of n photons, wherein n=0, 1, 2, .. . , n_(max); input each of values of P(n) as a n_(max)+1 component ofa feature vector into a single neuron neural network, said single neuronneural network having been previously trained on a plurality of lightsource types; and provide as output a classifier that has a value thatidentifies a light source type, wherein an average number of photons insaid plurality of time bins is less than one photon.
 11. The lightdetection system according to claim 9, wherein said light source type isone of a coherent light source or a thermal light source.
 12. The lightdetection system according to claim 11, wherein n_(max)=6 and saidfeature vector is a seven-component feature vector.
 13. The lightdetection system according to claim 12, wherein said single neuronneural network comprises an identity activation function and a binaryclassification given by a threshold function to indicate a class labeledas coherent on a first side of a threshold or a class labeled thermal ona second side of said threshold.
 14. The light detection systemaccording to claim 9, wherein said plurality of time bins is less than100.
 15. The light detection system according to claim 11, wherein saidplurality of time bins each have substantially equal temporal widths andhave a value selected to correspond to a coherence time of said coherentlight source.
 16. The optical imaging system according to claim 10,wherein said light source type is one of a coherent light source or athermal light source.
 17. The optical imaging system according to claim16, wherein n_(max)=6 and said feature vector is a seven-componentfeature vector.
 18. The optical imaging system according to claim 17,wherein said single neuron neural network comprises an identityactivation function and a binary classification given by a thresholdfunction to indicate a class labeled as coherent on a first side of athreshold or a class labeled thermal on a second side of said threshold.19. The optical imaging system according to claim 10, wherein saidplurality of time bins is less than
 100. 20. The optical imaging systemaccording to claim 16, wherein said plurality of time bins each havesubstantially equal temporal widths and have a value selected tocorrespond to a coherence time of said coherent light source.